Hoelle,m. Game Theory. Parduespr12

8/13/2019 Hoelle,m. Game Theory. Parduespr12

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Game Theory

Matthew HoellePurdue University

Department of Economics

Spring 2012


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Contents

I Static Games of Complete Information 1

1 A Beautiful Game 3

1.1 Introduction to Game Theory . . . . . . . . . . . . . . . . . . . . . . . . . . 31.2 Motivating Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.3 The Normal Form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71.4 Dominant Strategies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91.5 Iterated Deletion of Strictly Dominated Strategies . . . . . . . . . . . . . . . 111.6 Nash Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

1.6.1 Collective Action Problems . . . . . . . . . . . . . . . . . . . . . . . 161.6.2 Coordination Games . . . . . . . . . . . . . . . . . . . . . . . . . . . 181.6.3 Anti-coordination Games . . . . . . . . . . . . . . . . . . . . . . . . . 201.6.4 Random Strategy Games . . . . . . . . . . . . . . . . . . . . . . . . . 21

1.7 Solution to the Motivating Example . . . . . . . . . . . . . . . . . . . . . . . 231.8 Application: Political Economy . . . . . . . . . . . . . . . . . . . . . . . . . 25

2 Lunch Trucks at the Beach 31

2.1 Motivating Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 312.2 Hotelling Model of Spatial Competition . . . . . . . . . . . . . . . . . . . . . 322.3 Cournot Duopoly . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

2.4 Bertrand Duopoly . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 372.5 Solution to the Motivating Example . . . . . . . . . . . . . . . . . . . . . . . 40

3 One of the Classic Blunders 45

3.1 Motivating Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 453.2 Solving the Battle of Wits . . . . . . . . . . . . . . . . . . . . . . . . . . . . 473.3 Solving Similar Games . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

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3.4 Symmetric Mixed Strategies of the Room Selection Game . . . . . . . . . . . 533.5 Properties of Mixed Strategy Equilibria . . . . . . . . . . . . . . . . . . . . . 54

3.5.1 2x2 Games . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 543.5.2 Games with Larger Strategy Sets . . . . . . . . . . . . . . . . . . . . 56

II Dynamic Games of Complete Information 61

4 Know When to Hold Em and When to Fold Em 63

4.1 Motivating Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 634.2 Dening Contingent Actions . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

4.3 The Logic of Backward Induction . . . . . . . . . . . . . . . . . . . . . . . . 684.4 The Dynamic Room Selection Game . . . . . . . . . . . . . . . . . . . . . . 744.5 Entry Deterrence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 774.6 Stackelberg Duopoly . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 794.7 Application: Primaries and Runos . . . . . . . . . . . . . . . . . . . . . . . 81

4.7.1 Dynamic Voting Game . . . . . . . . . . . . . . . . . . . . . . . . . . 814.7.2 "You Cant Handle the Truth" . . . . . . . . . . . . . . . . . . . . . . 84

5 The Strategy of Law and Order 89

5.1 Motivating Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 895.2 Information Sets and Subgames . . . . . . . . . . . . . . . . . . . . . . . . . 895.3 Relation to the Normal Form . . . . . . . . . . . . . . . . . . . . . . . . . . 925.4 Finitely Repeated Games . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 945.5 The Plea Bargain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 995.6 Solution to the Motivating Example . . . . . . . . . . . . . . . . . . . . . . . 102

6 Cooperation through Repeated Interaction 107

6.1 Motivating Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1076.2 Discounting Future Payos . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1086.3 Innitely Repeated Prisoners Dilemma . . . . . . . . . . . . . . . . . . . . . 109

6.3.1 Grim Trigger . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1106.3.2 Tit-for-Tat . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1146.3.3 k-Period Punishment . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

6.4 Rubinstein Bargaining . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122


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III Games of Incomplete Information 125

7 The Theory of Auctions 127

7.1 Motivating Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1277.2 Types of Auctions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1287.3 Second Price Auction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1287.4 First Price Auction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131

7.4.1 Complete Information . . . . . . . . . . . . . . . . . . . . . . . . . . 1317.4.2 Incomplete Information . . . . . . . . . . . . . . . . . . . . . . . . . . 131

7.5 Revenue Equivalence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140

8 Forming Beliefs ahead of a Date 1438.1 Motivating Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1438.2 Information and Beliefs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1448.3 Solution to the Motivating Example . . . . . . . . . . . . . . . . . . . . . . . 150

9 Beer or Quiche? 155

9.1 Motivating Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1559.2 Adverse Selection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1569.3 Signaling I: Beer-Quiche . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157

9.4 Signaling II: Job Market . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1649.4.1 Complete Information . . . . . . . . . . . . . . . . . . . . . . . . . . 1669.4.2 Asymmetric Information . . . . . . . . . . . . . . . . . . . . . . . . . 166

9.5 Screening: Insurance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1699.5.1 Complete Information . . . . . . . . . . . . . . . . . . . . . . . . . . 1709.5.2 Asymmetric Information . . . . . . . . . . . . . . . . . . . . . . . . . 172

10 Writing Incentives into Contracts 177

10.1 Motivating Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17710.2 Kuhn-Tucker Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17810.3 Principal-Agent Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17810.4 Complete Information . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17910.5 Asymmetric Information . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18110.6 Algebraic Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186

10.6.1 Maximization Problem . . . . . . . . . . . . . . . . . . . . . . . . . . 186


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10.6.2 Both constraints bind . . . . . . . . . . . . . . . . . . . . . . . . . . . 18710.6.3 Lambda . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187

10.6.4 Mu . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188

IV Solutions to Selected Exercises 189

11 Chapter 1 Solutions 191











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Preface

These notes have been prepared for a 40-hour undergraduate game theory course. Thiscourse is designed for students without any prior familiarity with game theory. However,it is advised that students have the appropriate mathematical background, specically theability to calculate derivatives and partial derivatives. To this end, a list of mathematicalpreliminaries is included at the end of the Preface.

The notes contain a large number of exercises to be completed outside of lecture. Theexercises vary in diculty, and each has a solution provided in the last part of the manuscript.It is advised that students make an honest attempt to solve the exercises, before giving upand looking to the solutions for guidance. The expectation is that the amount of out-of-lecture learning and exercise solving accounts for approximately 120 hours (three times theamount of time spent in lecture). This practice is required for a student to fully understand

the game theory concepts and to gain the maturity necessary to apply these concepts toreal-life applications.

The notes are divided into three parts, with a fourth part for the solutions to the exercises.Part I considers Static Games of Complete Information. Within Part I, Chapter 1 con-

siders discrete choice games (pure strategies), Chapter 2 considers continuous choice games,and Chapter 3 considers the mixed strategies of discrete choice games.

Part II considers Dynamic Games of Complete Information. Within Part II, Chapter 4considers games of perfect information, Chapter 5 considers games of imperfect information,and Chapter 6 considers innitely repeated games.

Part III considers Games of Incomplete Information. Within Part III, Chapter 7 considersauctions, Chapter 8 considers the Perfect Bayesian Nash Equilibrium concept, Chapter 9considers adverse selection, and Chapter 10 considers moral hazard.

Each exercise contained in the course of the rst 10 chapters has a solution that iscontained in Part IV. Part IV includes Chapters 11-20, where the chapters correspond toChapters 1-10.

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Within the early chapters, I have included several "problems." These problems will bethe focus of classroom discussion. As prizes will be awarded to those students that play the

game the "best," I do not include solutions to these "problems."I would like to thank Somak Paul and the Fall 2011 class of Econ 451 at Purdue University.

They essentially served as guinea pigs for this material that I have developed. Their questionsand passion for the material greatly helped me improve the presentation in this text.

Mathematical Preliminaries The following list provides the key mathematical resultsthat are required to solve problems in this class.

1. First derivatives

(a) ddx (cx) = c

(b) ddx (cxn ) = cnx n 1 for n 6= 0

(c) ddx (c ln(x)) = cx for x > 0

(d) Chain Rule: If f (x) = h (g(x)) ; then ddx f (x) = ddy h (y)

ddx g(x); where y = g(x):

(e) Product Rule: ddx (f (x) g(x)) = ddx f (x) g(x) + f (x)

ddx g(x):

(f) Quotient Rule: ddxf (x )g(x ) =

ddx f (x ) g(x ) f (x )

ddx g(x )

[g(x )]2 :

2. Partial derivatives

(a) @ @xf (x; y ) = ddx f (x ; y); where y is held xed and held constant. For example,

@ @x (x(y x)) = y 2x:

3. Property of the natural log

(a) ln(a) + ln( b) = ln( ab):

4. Summation notation

(a)n

Pi=1 x i = x1 + x2 + ::: + xn :5. Set notation

(a) The set X = fIndianapolis; Chicago; Detroit gis a set containing three elements.


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CONTENTS ix

(b) The set X nfDetroit g equals the set X dened above, but excluding the elementDetroit. The set X nfDetroit g= fIndianapolis; Chicago g:

6. Vector notation

(a) If xi is a one-dimensional variable and i = 1 ;:::;I; then x = ( x1;:::;x I ) is anI dimensional vector.


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Part I

Static Games of Complete

Information

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Chapter 1

A Beautiful Game

1.1 Introduction to Game Theory

It is no coincidence that the scope for economic analysis has expanded in the past 60 yearsin unison with the growth and applicability of game theoretic reasoning. Where once eco-nomics was adequately dened as the study of market prices and their eects on individualdecision-making, economics as it stands today requires a broader denition. My favoriteversion denes economics as the study of incentives in all social institutions.

Consider the interaction of individuals in the classical economic theory of markets. In-dividuals trade on competitive markets, meaning that they take prices as given and do notconsider that either their own actions, or the actions of other individuals in the market, canresult in dierent terms of trade. While this competitive interaction of individuals is still agood approximation for "large" markets, it is clear that many examples in life involve fewerindividuals. In this case, the individuals have market power, so we want to be able to saysomething about the strategic interaction of individuals. Game theory provides the toolsthat allow us to predict outcomes in settings of strategic interaction.

The following are just a few real-life examples of strategic interaction:

eBay auctions

college admissions

voting

military standos

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4 CHAPTER 1. A BEAUTIFUL GAME

marketing and advertising

soccer penalty kicks

marriage markets

student movie discounts

health insurance

contract enforcement

bargaining and negotiation

To understand game theory, we must rst understand the meaning of a game. A gamehas a specic denition given by economists. It is important not to limit our thinking abouta game to its colloquial usage. While board games, card games, and games of chances arecertainly examples of games, they are but a small subset of the possible range of gamesthat can be addressed using the logic of game theory.

To appropriately dene a game, lets consider that you and your friend decide to passthe time by playing a board game. You open the closet in your dorm room and take downyour favorite board game. The problem is that your friend has never played this particulargame before. So you open the box and the rst item that you should see is the piece of paperthat details the rules of the game. A game is dened according to its rules. These rulesmust be written so that your friend, who has never played the game before, can read themand completely understand how the game is played.

Lets break down the rules of a game into four components. These components are thefour characteristics that we use to dene a game.

1. How many players can play the game (depending on the context throughout this text,

I will use the terms agents, households, decision-makers, and rms to describe theindividuals that play the game)

2. What is the objective of players (in game theory, the objective of players is to maxi-mize their payo functions, where the payo functions can be something as simple as"maximize the probability that you are the rst player to successfully move all 4 of your pawns from Start to Home")


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We shall later see that the prediction described in the lm is incorrect, using the theory of strategic interaction that was introduced by John Nash (and later coined Nash equilibrium).

Nonetheless, the point that can be taken from this scene is that players need to make theirdecisions by taking into account the actions of other players in the game.

For our purposes, lets consider an equivalent and simpler game called the Room SelectionGame. In this game, three female roommates are deciding how to allocate three bedrooms ina shared apartment. The rooms have dierent features and all roommates agree that Room1 is the best, Room 2 is the second-best, and Room 3 is the worst. We suppose that therooms are allocated according to the following game. All three roommates write down theirrankings over the three rooms. These rankings are private, meaning that a player knowsher own ranking, but not the rankings of the others. The rankings are then collected by anadministrator.

The rankings are revealed by the administrator. If a player is the only roommate to lista particular room as her rst choice, then she receives that room with probability 100%. If a player is one of two roommates to list a particular room as her rst choice, then a coinis ipped and each has a 50% probability of receiving the room. If all three players list thesame room as their rst choice, then each has a 33.33% probability of receiving the room.

After the allocation of the rooms that are rst choices, the remaining rooms are thenallocated based upon the second and third choices of players. For example, suppose that

two roommates submitted the rankings Room 1 - Room 2 - Room 3, while one roommatesubmitted the rankings Room 2 - Room 1 - Room 3. What happens to the roommate thatloses the coin ip for Room 1? Well, that player cannot receive her second choice, Room2, as this room has already been allocated. Thus, the roommate that loses the coin ip forRoom 1 will receive Room 3.

If you were one of the roommates playing this game, what ranking would you write down?

More importantly, how do we predict that this game will be played? We predict that thegame will be played using Nash equilibrium strategies. An equilibrium, as in chemistry, is asystem at rest.

For this game, the system is the following hypothetical convergence process (the processis hypothetical, because all players have prior knowledge about the other players payosand can accurately forecast how each of them will form strategies). Each of the players (insecret) writes down her ranking over the three rooms. All rankings are then placed on a tablein the middle of the three players and revealed to all three. The administrator then selectsa player at random. That player is given the opportunity, given what she now knows about


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1.3. THE NORMAL FORM 7

the strategies of the other players, to change her strategy. If the ranking is changed, thenthe updated ranking is placed on the table for all players to see. The administrator then

selects a dierent player at random and gives that player the same opportunity to changeher ranking. The system is at rest (an equilibrium exists) when the administrator asks eachof the three players in turn if she wants to change her ranking, and each one declines theoer.

What then is the equilibrium for this particular game? That analysis is delayed untilafter the game theory concepts have been introduced.

1.3 The Normal Form

The games considered in this chapter are static games. This simply means that all playersmake simultaneous decisions and then the game ends. What is important for the concept of simultaneous choice is not the timing, but rather the information possessed by the players.The players do not actually have to make their decisions simultaneously. Rather, each playeronly needs to make its decision without knowing what decisions have been made by the otherplayers. Consider the Room Selection Game in which the roommates write down rankingsof the rooms in secret. The roommates need not make these decisions at the exact sameinstant in time, but the rankings must be secret. This means that each player does not have

knowledge of the actions of the other players when called upon to make its decision.The normal form is a convenient way to depict each of the four characteristics of a

game (players, payos, strategy sets, and eects on other players). The normal form is alsocommonly called the matrix form. Consider the following example.

Column PlayerL C R

Row U 2, 2 1, 4 4, 3Player D 1, 1 0, 2 3, 1

Example 1.1

Example 1.1 describes a game played between two players: Row Player and ColumnPlayer. The Row Player can either choose the strategy U or the strategy D (think Up andDown), while the Column Player can choose the strategy L, C, or R (think Left, Center,and Right). The payos for each of the players is depicted as elements in the 2 3 matrix.The convention is that the payo for the Row Player is listed rst, followed by a comma,


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followed by the payo for the Column Player. For example, if Row Player chooses U andColumn Player chooses C, then the outcome is (U,C) with a payo of 1 for Row Player and

4 for Column Player.As a second example, I will introduce the story behind possibly the most famous example

in all of game theory: the Prisoners Dilemma. Two male prisoners are arrested on suspicionof having jointly committed a large crime and a small crime. They are interrogated inseparate rooms. The police only have evidence to convict each of the prisoners for committingthe small crime. This is known by the prisoners.

Given this bind on the part of the police, they make the following proposal to Prisoner1. If Prisoner 1 informs against Prisoner 2, then the police will have enough evidence toconvict Prisoner 2 of committing the large crime. As a reward for cooperation, if Prisoner 1informs against Prisoner 2, the police will then drop the small crime charge against Prisoner1. The exact same proposal is made to Prisoner 2.

Suppose that a small crime conviction requires 1 year of prison time and a large crimeconviction requires 3 years of prison time. The reward for cooperation is equal to a 1 yearreduction in prison time (equivalent to dropping the small crime charges).

Notice that if both prisoners remain silent, then each of them will be convicted of the smallcrime and sentenced to 1 year in prison. If both prisoners inform against their counterpart,they are both convicted of the small and large crime (1+3=4 year prison sentence), but arerewarded with a 1 year prison reduction for having cooperated with the police.

This game is depicted in the normal form as follows.

Suspect 2Mum Fink

Suspect Mum -1, -1 -4, 01 Fink 0, -4 -3, -3

Example 1.2

The game is old (circa 1950s), so the dierent strategies are given old-fashioned names.Mum means that a prisoner remains silent, while Fink means that a prisoner informsagainst his counterpart. The payos listed in the 2 2 matrix reect the story describedabove, where the absolute values of the payos are the lengths of the prison sentences andwe suppose that less time in prison is better than more.

The third example of the normal form is given below.


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1.4. DOMINANT STRATEGIES 9

Column PlayerL C R

Row U 2, 2 1, 4 3, 3Player D 1, 3 0, 2 4, 1

Example 1.3

The fourth example of the normal form is given below.

Column PlayerL C R

Row U 2, 2 1, 4 3, 3

Player D 1, 3 0, 1 4, 2Example 1.4

1.4 Dominant Strategies

We, as economists, want to be able to make predictions about how the games given inExamples 1.1, 1.2, 1.3, and 1.4 will be played. We begin with the rst logical rule: all playerswill play their dominant strategy, if they have one. What is a dominant strategy? It is a

strategy that provides the weakly highest payo for a particular player, no matter whatstrategies are employed by the other players.

Mathematically, we dene a dominant strategy using the following notation. For a gamewith I players, let pi (s) be the payo to player i as a function of all players strategies:

s = ( s1; s 2;:::;s i ;:::;s I ) :

For simplicity, dene s i as the strategies for all players, except player i :

s i = ( s1; s 2;:::;s i 1; s i+1 ;:::;s I ) :

The strategy si is dominant for player i if for any s i and any alternative strategy ~s i :

pi (s i ; s i ) pi (~s i ; s i ) :

Lets consider some examples of dominant strategies. From Example 1, lets focus only


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on the payos for the Row Player:

Column PlayerL C R

Row U 2, ... 1, ... 4, ...Player D 1, ... 0, ... 3, ...

.

In doing so, we can see that the strategy U always provides a higher payo than the strategyD, no matter what strategy is chosen by the Column Player (if Column Player chooses L,then U provides 2 and D only provides 1; if Column Player chooses C, then U provides 1and D only provides 0; if Column Player chooses R, then U provides 4 and D only provides3). Thus, U is a dominant strategy for the Row Player. Now, lets focus only on the payosfor the Column Player:

Column PlayerL C R

Row U ..., 2 ..., 4 ..., 3Player D ..., 1 ..., 2 ..., 1

.

As can be seen, the Column Player also has a dominant strategy, which is to play C.

Each of the two players in Example 1.1 has a dominant strategy. Thus, we predict thatthe outcome of the game will have each of the players choosing this dominant strategy, sothe outcome will be (U,C).

From Example 1.2, lets focus only on the payos for Prisoner 1:

Suspect 2Mum Fink

Suspect Mum -1, ... -4, ...1 Fink 0, ... -3 , ...

.

Prisoner 1 has a dominant strategy, which is Fink. This particular game is symmetric. Asymmetric game is one in which ipping the actions of the rst and the second player resultsin a ipping of the payos. For this reason, if Fink is a dominant strategy for Prisoner1, then Fink is a dominant strategy for Prisoner 2. If you dont believe me, consider the


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1.5. ITERATED DELETION OF STRICTLY DOMINATED STRATEGIES 11

following matrix in which we only focus on the payos for Prisoner 2:

Suspect 2Mum Fink

Suspect Mum ..., -1 ..., 01 Fink ..., -4 ..., -3

.

What about Example 1.3? For this example, when we focus on the payos of the RowPlayer

Column PlayerL C R

Row U 2, ... 1, ... 3, ...Player D 1, ... 0, ... 4, ...

we cannot nd a dominant strategy. In this case, U is a better choice if the Column Playerchooses L or C, but not if the Column Player chooses R. Similarly, if we focus on the payosof the Column Player

Column PlayerL C R

Row U ..., 2 ..., 4 ..., 3Player D ..., 3 ..., 2 ..., 1

we also fail to nd a dominant strategy. This suggests that we would like to have a morepowerful tool to be able to make predictions not only in the games of Example 1.1 and 1.2,but in the game in Example 1.3 as well.

1.5 Iterated Deletion of Strictly Dominated Strategies

To introduce this new concept of Iterated Deletion of Strictly Dominated Strategies (ab-breviated as IDSDS), we rst dene Common Knowledge. The assumption of CommonKnowledge states that the Row Player knows the payos for the Column Player and canlogically deduce its course of action. The Column Player knows that the Row Player hasthis information. The Row Player knows that the Column Player knows that the Row Playerhas this information. This higher order knowledge continues without end.

The opposite of a dominant strategy is a dominated strategy. A strategy is said tobe strictly dominated if it provides strictly lower payos than another strategy, no matter


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the strategies of the other players. Mathematically, using the denitions from the previoussection, the strategy s i is strictly dominated by ~s i for player i if for any s i :

pi (~s i ; s i ) > p i (s i ; s i ) :

The logic of IDSDS begins with the realization that a player will never play a strictlydominated strategy. Consider the process of IDSDS as applied to Example 1.3. Focusing onthe payos for the Column Player from playing C and R,

Column PlayerL C R

Row U ..., xx ..., 4 ..., 3Player D ..., xx ..., 2 ..., 1

,

we see that the Column Player will never select R (it is strictly dominated by C). The RowPlayer knows this (Common Knowledge), so does not take into account the possible outcomes(U,R) and (D,R):

Column PlayerL C R

Row U 2, 2 1, 4 XXXX

Player D 1, 3 0, 2 XXXX

.

With the game trimmed in the above manner, focus on the payos for the Row Player:

Column PlayerL C R

Row U 2, ... 1, ... XXXXPlayer D 1, ... 0, ... XXXX

.

The strategy D is strictly dominated and will not be played by the Row Player. The Col-

umn Player knows this (Common Knowledge) and does not take into account the possibleoutcomes (D, L) and (D, C):

Column PlayerL C R

Row U 2, 2 1, 4 XXXXPlayer D XXXX XXXX XXXX


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1.6. NASH EQUILIBRIUM 13

With the game logically updated for the Column Player as above, its strategy of L is strictlydominated by C. Thus, the logic of IDSDS dictates that outcome is (U, C).

Problem 1.1 Lets see how the logic of IDSDS applies to a Bizzaro "Price is Right" problem.Suppose that you are one of four contestants asked to bid on a Gizmo. You have no idea what the price of a Gizmo should be (as it is an imaginary object), but Drew Carey informs you that the price of a Gizmo is equal to 75% of the averge bids of all four contestants. Drew then asks all contestants to write down a bid for a Gizmo (in secret) between $1 and $1,000.The contestant with the bid closest to the actual price will win.

You must write down a bid. The fact that the bid is made in secret means that you must write down your bid before observing what your three fellow contestants have bid. What is

the bid that you will write down? If you think that this bid will win, what are you hoping about the deductive abilities of your fellow contestants?

Notice that the logic of IDSDS can be used to obtain a unique prediction in the gamesin both Examples 1.1 and 1.2. Thus, IDSDS is more general than dominant strategies as itallows for a prediction in all games that dominant strategies does, in addition to allowingfor a prediction in the game of Example 1.3. Is this the most general method that we canemploy? Lets see if the method can be applied to Example 1.4.

Column PlayerL C R

Row U 2, 2 1, 4 3, 3Player D 1, 3 0, 1 4, 2

You can stare at Example 1.4 all day, but you wont be able to nd a strategy that isstrictly dominated for either player. This means that the IDSDS method cannot even begin.We seek a method that allows us to form predictions in the games in all Examples 1.1, 1.2,1.3, and 1.4.

1.6 Nash Equilibrium

The concept of a Nash equilibrium is credited to the groundbreaking papers of John Nashthat are alluded to in the movie A Beautiful Mind. The concept of a Nash equilibriumfollows the fundamental principle of a system at rest. Specically, a Nash equilibrium is the


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set of strategies for all players such that each player is playing a best response to every otherplayers best response. Mathematically, using the notation introduced two sections prior, a

Nash equilibrium iss = ( s1; s 2;:::;s i ;:::;s I )

with the property that for each player i :

pi s i ; s i pi s i ; s i

for all other possible strategies s i :This equilibrium concept is more general than both dominant strategies and IDSDS. The

following facts collect the logic.

1. If a player has a dominant strategy, any Nash equilibrium must contain this dominantstrategy.

2. If the process of IDSDS results in a unique outcome (unique strategies for all players),then those strategies must be a Nash equilibrium. Further, this is the only Nashequilibrium.

The Nash equilibria of any two-player discrete choice game are found using the underline

method. Here is how the underline method works applied to a 3 3 normal form game.

Column PlayerL C R

T 3; 3! 1; 5 3; Row Player M 5; 52 4; e 4; p 8

B 2; 1 5; 1 1; 0Example 1.5

.

Begin with the top row. Underline the largest payo value for the Column Player.

Column PlayerL C R

T 3; 3! 1; 5 3; Row Player M 5; 52 4; e 4; p 8

B 2; 1 5; 1 1; 0

.


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Proceed to the second and third row. In each case, underline the largest payo value for theColumn Player.

Column PlayerL C R

T 3; 3! 1; 5 3; Row Player M 5; 52 4; e 4; p 8

B 2; 1 5; 1 1; 0

.

In the case of a tie, as in the third row in which 1 is the highest value and appears in twocells, then you need to underline each instance in which the highest value appears.

To help you avoid mistakes when using the underline method, just remember that thelogic is backwards. When looking across a row , you are underlining the highest payo forthe Column Player (row > Column Player ).

The underline method continues by looking at the columns. For each column, underlinethe largest payo value for the Row Player ( column > Row Player ).

Column PlayerL C R

T 3; 3! 1; 5 3; Row Player M 5; 52 4; e 4; p 8

B 2; 1 5; 1 1; 0

.

Any cell that contains two underlines is a Nash equilibrium. A Nash equilibrium is the setof strategies, while Nash equilibrium payos are the two payo values. For Example 1.5, thetwo Nash equilibria are (M,R) and (B,C), with payos of 4; p 8 and (5; 1) ; respectively.

Why does the underline method work (and it always works to nd all Nash equilibriain any two-player discrete choice game)? What we are underlining are actually the bestresponses for the players. As a Nash equilibrium is a best response to a best response, thentwo underlines yields a Nash equilibrium.

For the games in Examples 1.1, 1.3, and 1.4, the Nash equilibrium in all cases is (U,C),


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where the underline method for each is given below.

Column PlayerL C R

Row U 2, 2 1, 4 4, 3Player D 1, 1 0, 2 3, 1

Example 1.1

Column PlayerL C R

Row U 2, 2 1, 4 3, 3Player D 1, 3 0, 2 4, 1

Example 1.3

Column PlayerL C R

Row U 2, 2 1, 4 3, 3Player D 1, 3 0, 1 4, 2

Example 1.4

Exercise 1.1 Find the Nash equilibria of the following normal form game:

Player 2 L C R

T 3, 2 1, 6 3, 2 Player 1 M 5, 1 4, 3 3, 2

B 2, 4 5, 4 1, 4

Exercise 1.2 Find the Nash equilibria of the following normal form game:

Player 2 Save Use Delay

Save 5, 5 -2, 10 0, 0 Player 1 Use 3, 2 1, -1 0, 0

Delay 0, 0 0, 0 0, 0

The discrete choice games can be divided into four classes of games, where games withina particular class have similar predicted outcomes. The classes of games are (i) CollectiveAction Problems, (ii) Coordination Games, (iii) Anti-coordination Games, and (iv) RandomStrategy Games.

1.6.1 Collective Action Problems

This class of games is characterized by the following property: the unique Nash equilibriumis not Pareto optimal. The leading example of a collective action problem is the Prisoners


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Dilemma introduced in Example 1.2. Recalling the normal form game,

Suspect 2Mum Fink

Suspect Mum -1, -1 -4, 01 Fink 0, -4 -3, -3

,

the Nash equilibrium is (Fink, Fink). The equilibrium payos of ( 3; 3) are lower for bothsuspects compared to the outcome (Mum, Mum) with payos ( 1; 1) :

A second example of a collective action problem is the Tragedy of the Commons. In thisexample, consider a public plot of land that two farmers may use for sheep grazing. If the

resource is appropriately used, then grass is available for the sheep to eat, without a cost tothe farmer. If the resource is overused, then not enough grass can be grown to feed all thesheep. Dene B as the payo to a farmer if he is the only one of the two farmers that allowsits sheep to graze on the public land. Dene A as the payo to the farmer if neither of thetwo allow their sheep to graze on the public land. Dene a as the payo to the farmer if bothallow their sheep to graze on the public land. Finally, dene b as the payo to the farmerif he does not allow his sheep to graze on the public land, while the other farmer does. Thelogic of the problem dictates that

B > A > a > b:

The farmer prefers to be the only one using the public land. Provided that both farmers takethe same action, each would prefer to leave the land fallow and allow grass to be availablenext period, rather than consuming the entire resource in the current period.

In the normal form, the game is depicted as follows.

Farmer 2Sheep graze Sheep not graze

Farmer Sheep graze a, a B, b1 Sheep not graze b, B A, A

Example 1.6

Given the relation between payos, the underline method dictates that the unique Nashequilibrium is (Sheep graze, Sheep graze). The equilibrium payos of (a; a ) are strictly lowerfor both players than the payos for the outcome (Sheep not graze, Sheep not graze). This


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is the tragedy, which occurs because the players are unable to commit to a collective action.Each player acts in his best interest, with the result being a suboptimal outcome.

1.6.2 Coordination Games

The two leading examples of coordination games are the Battle of the Sexes and the StagHunt. Here is the background for the Battle of the Sexes. A Man leaves his oce in a worldwithout cell phones and e-mail and must decide to go either to the Opera or to a Boxingmatch. Meanwhile, a Woman across town leaves her oce at the same time and must decideto go either to the Opera or to a Boxing match. The Man prefers Boxing over Opera, whilethe Woman prefers Opera over Boxing, but the overarching concern of each is to be at thesame event together.

Assigning reasonable payos to the two players, the normal form of this game can bedepicted as follows.

WomanOpera Boxing

Man Opera 40, 60 0, 0Boxing 0, 0 70, 30

Example 1.7

Using the underline method, the two Nash equilibria of this game are (Opera, Opera) and(Boxing, Boxing).

Lets now consider the background for the Stag Hunt game. Hunter 1 wakes up in hiscastle in a world without cell phones and e-mail and must decide to either ride to the staghunting grounds or the hare hunting grounds (a stag is a large deer). At the same time,

Hunter 2 wakes up in a distant castle and must decide whether to ride to the same staghunting grounds or to the same hare hunting grounds. The nice thing about hunting a hareis that it requires only one hunter to complete the task. Of course, a hare only has so muchmeat. The larger animal, the stag, is harder to hunt and requires the combined eorts of both hunters.

Assigning reasonable payos to the two players, the normal form of this game is given


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below.Hunter 2Stag Hare

Hunter 1 Stag 2, 2 0, 1Hare 1, 0 1, 1

Example 1.8

Using the underline method, the two Nash equilibria of this game are (Stag, Stag) and (Hare,Hare).

This allows us to make the following two claims about the predicted play in CoordinationGames.

1. There are multiple Nash equilibria.

2. In the case of Stag Hunt, some equilibria are "better" than others, while other equilibriaprovide a risk-free payo. Notice that (Stag, Stag) provides strictly higher payos forboth hunters compared to (Hare, Hare). However, choosing Hare provides a guaranteedsafe payo of 1 for a hunter. For this reason, the Stag Hunt game is often called theAssurance Game.

Problem 1.2 Your friend is returning from her study abroad in Italy. The two of you have agreed to meet as soon as she returns, but given the poor quality of Italian internet, you have been unable to reach your friend for the past several weeks. All you know is that your friend will be on campus on January 8. You must pick one location and one time as the window tomeet your friend. If she picks the exact same location and the same time, then you two will meet up and both are happy. If the (location,time) pairs do not match, then you are unable to meet up.

Where would you choose to meet your friend? At what time? Suppose the stakes were raised. You are playing this game with a complete stranger, yet a student at Purdue. The administrator of the game asks you to choose one location and one time. If the other player

(the stranger) chooses the exact same time and location, then the administrator provides each of you with $100,000. If you and the stranger are unable to meet up, then both of you walk away empty handed. Do the higher stakes change your thinking about the game (or simply cause you to consider it more carefully)?

In this example of a coordination game, there are a large number (yet nite) of dierentstrategies that you can choose. You can choose to be at the Rawls bistro at 8 am, at Rawls


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bistro at 9 am, and so forth; at the bookstore at 8 am, at the bookstore at 9 am, and soforth; at the bell tower at 8 am, at the bell tower at 9 am, and so forth. The payos for

this game are 100,000 if both players choose the same strategy, and 0 otherwise. Using theunderline method, we can see that for each (time, location) pair that we can imagine, a Nashequilibrium exists in which both players choose that particular pair.

This predicts a large number of Nash equilibria. Are some more likely than others?Lets use the concept of a Schelling focal point to address this question. Thomas Schellingsuggested that some of the Nash equilibria are focal in the minds of the players. Focal simplymeans the human mind is quicker to concentrate on that particular outcome. The playersare then more likely to play a strategy that is focal. What would you say is a focal time andlocation on Purdues campus? A correct answer could net $100,000.

The concept of a focal point relates to the idea of a social norm. A social norm is behaviorthat has been adopted by society as "appropriate" without the use of laws to coordinate allindividuals. Examples include walking on the right hand side of sidewalks in a U.S. cityor clapping at the end of a theatre play (but not at the end of a lecture). Consider thesidewalk example. The action that any one individual takes must be coordinated with theactions of all the other individuals in the city. Otherwise, if I walk on the left hand sideand everyone else walks on the right hand side, then my travel is slowed as I am constantlyrunning into people. Thus, it is clearly a Nash equilibrium for all individuals in a city to

walk on the right hand side of sidewalks. The interesting thing is that this is not the onlyequilibrium. Another equilibrium would be for all individuals to walk on the right handside on even-numbered avenues and on the left hand side on odd-numbered avenues. Thisis also an equilibrium as it prevents people from running into each other, but it is terriblydicult to remember. A social norm is an equilibrium that is easy to remember and apply.In essence, a social norm is a Schelling focal point.

1.6.3 Anti-coordination Games

The example of an anti-coordination is the game of Chicken. In this game, two drivers aredriving cars toward each other at high rates of speed. Right before the instant of collision,each of the drivers can choose to either swerve to the left or go straight. If both choose to gostraight, then they will crash, with this outcome resulting in the lowest possible payos forboth drivers. If one swerves while the other goes straight, then the driver that has swerved


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is the chicken and is mocked by his peers. The game can be depicted below.

Driver 2Straight Swerve

Driver 1 Straight 0, 0 10, 1Swerve 1, 10 5, 5

Example 1.9

Using the underline method, there are two Nash equilibria of the Chicken game: (Straight,Swerve) and (Swerve, Straight).

1.6.4 Random Strategy GamesThe examples of random strategy games are given by simple childrens games. The rst iscalled Matching Pennies. Each player has a penny. On the count of 3, each of the playersmust either show the penny heads side up or tails side up. Player 1 wins the game if thefaces of the two pennies match: two heads face up or two tails. Player 2 wins the game if the faces of the pennies do not match. Assigning the value of 1 for a win and -1 for a loss,the game is depicted in the normal form below.

Player 2Heads Tails

Player 1 Heads 1, -1 -1, 1Tails -1, 1 1, -1

Example 1.10

Using the underline method, there are no Nash equilibria of this game.Examine the logic for why a Nash equilibrium cannot exist. Suppose Player 1 plays

Heads. Then the best thing that Player 2 can do is play Tails (no match). Well when Player2 plays Tails, the best thing that Player 1 can do is play Tails (match). Thus, there neverexists a system at rest in which both players are happy with their current strategy and itsresulting payo.

The second childrens game is Rock Paper Scissors, which is played between two players.In this game, after the signal "Rock, Paper, Scissors, Shoot" each player forms their righthand into the shape of a rock (a st), paper (a at hand), or scissors (index and middlenger extended). Logic dictates that rock crushes scissors, scissors cuts paper, and paper


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covers rock. If both players throw the same shape, then the game ends in a tie. Assigningthe value of 1 for a win, 0 for a tie, and -1 for a loss, the game is depicted below.

Player 2Rock Paper Scissors

Rock 0, 0 -1, 1 1, -1Player 1 Paper 1, -1 0, 0 -1, 1

Scissors -1, 1 1, -1 0, 0Example 1.11

Using the underline method, there are no Nash equilibria of this game.For complete disclosure, the Nash equilibria that we nd in this chapter are pure-strategy

Nash equilibria. Chapter 3 analyzes mixed-strategy Nash equilibria and variants of the gamesin Examples 1.10 and 1.11. A Nash equilibrium can be either a pure-strategy or a mixed-strategy Nash equilibrium. Theory dictates that a Nash equilibrium must exist. Thus, if apure-strategy Nash equilibrium does not exist (as in the games of Example 1.10 and 1.11),then a mixed-strategy Nash equilibrium must exist. Further details are postponed untilChapter 3.

Exercise 1.3 Write the following games in the normal form and solve for the Nash equilib-

ria. In some cases, multiple Nash equilibria may exist or a Nash equilibrium may not exist at all. (Note: There is some freedom in how you assign payo values for the players. These choices may aect the Nash equilibria.)

(a) Two animals are ghting over some prey (this game is typically called a Hawk-Dove game). Each can be passive or aggresive. Each prefers to be aggresive if its opponent is passive, and passive if its opponent is aggresive. Given its own stance, it prefers the outcome in which its opponent is passive to that in which its opponent is aggresive.

(b) Two students wish to attend the same university. The students each receive strictly positive payo if they attend the same university and zero payo otherwise. The list of possible universities is Purdue University, Indiana University, and Notre Dame University.Student A prefers Purdue to Indiana and Indiana to Notre Dame (by transitivity, he/she also prefers Purdue to Notre Dame). Student B has a scholarship at Notre Dame, so prefers Notre Dame to Purdue and Purdue to Indiana (by transitivity, he/she prefers Notre Dame to Indiana).

(c) Consider a soccer penalty kick between a Scorer and a Goalie. The Scorer is stronger


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1.7. SOLUTION TO THE MOTIVATING EXAMPLE 23

kicking to his/her Right than to his/her Left. Given the 12 yards between the ball and the goal, the Goalie must pre-determine which way he/she will dive. Thus, the actions are chosen

simultaneously: a Scorer can shoot either Left or Right and the Goalie can dive either Left or Right. If the Goalie guesses wrong, the Scorer always scores. If the Goalie guesses correctly with the Scorer kicking Right, then 50% of the shots are stopped. If the Goalie guesses correctly with the Scorer kicking Left, then 90% of the shots are stopped.

1.7 Solution to the Motivating Example

Recall that three roommates are submitting rankings over three bedrooms (Room 1, Room2, Room 3) in a shared apartment. The three rooms (Room 1, Room 2, Room 3) have valuesequal to (a;b;c) = (30 ; 24; 6) ; respectively. Each of the roommates must decide which of thepossible rankings to submit. The 3! = 6 possible rankings are f123; 132; 213; 231; 312; 321g:It is not optimal for a roommate to list Room 3 anywhere but as her last choice. Considerthat by listing Room 3 as either her rst or second choice, a roommate is forfeiting theopportunity to enter a lottery with the chance to earn a strictly higher payo. Thus, theonly two possible rankings that will be chosen in equilibrium are f123; 213g:

There are four possible outcomes that can be attained for varying combinations of thestrategies chosen by the three roommates.

1. If all three roommates choose 123; then each has a 13 chance of winning its rst choice(Room 1). Each has a 23 chance of losing its rst choice. Given that only two roommatesremain for the two rooms, each would then have a 12 chance of winning its second choice(Room 2). This means that the ex-ante chance of winning the second choice (Room 2)is 23 12 = 13 : If a roommate loses both the lotteries for Room 1 and Room 2, then it isassigned the lone remaining room (Room 3). The chance of this outcome is 13 : Thus,the expected payo for each roommate is 13 (30) +

13 (24) +

13 (6) = 20 :

2. If two roommates choose 123 and one chooses 213; then the lone roommate that chooses

213 will receive Room 2 with certainty (and a payo of 24). Each of the roommatesthat chooses 123 has a 12 chance of winning Room 1 (its rst choice) and a

12 chance

of winning Room 3 (the lone remaining choice). Thus, the expected payo for each of the roommates that chooses 123 is 12 (30) +

12 (6) = 18 :

3. If one roommate chooses 123 and two choose 213; then the lone roommate that chooses123 will receive Room 1 with certainty (and a payo of 30). Each of the roommates


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that chooses 213 has a 12 chance of winning Room 2 (its rst choice) and a 12 chance

of winning Room 3 (the lone remaining choice). Thus, the expected payo for each of

the roommates that chooses 213 is 12 (24) +

12 (6) = 15 :

4. If all three roommates choose 213; then the same logic as Case 1 implies that theexpected payo of each is 13 (30) +

13 (24) +

13 (6) = 20 :

Remember that a Nash equilibrium is the set of three strategies (rankings) such thateach roommate is playing a best response to the other players best responses. Is it thenpossible that (123; 123; 123) is a Nash equilibrium (namely, all roommates choose the sameranking 123)? The payo outcome is 20 for all three. Can one of the players (holding xedthe strategies of the other two) change its strategy and receive a higher expected payo?Certainly. Consider that a roommate can receive a payo value of 24 by using the ranking213:

So if (123; 123; 123) is not a Nash equilibrium, how can we nd the Nash equilibria?The normal form game is a 2 2 2 array. To depict this 3-dimensional array in 2-dimensional space, we consider the following two cross-sections. In the rst, we hold xedPlayer 1s strategy at 123: In the second, we hold xed Player 2s strategy at 213: The cellsof the matrices contain three payo values: the rst belongs to roommate 1, the second toroommate 2, and the third to roommate 3.

1 plays 123 Roommate 3123 213

Roommate 2 123 20, 20, 20 18, 18, 24213 18, 24, 18 30, 15, 15


Roommate 2 123 24, 18, 18 15, 30, 15213 15, 15, 30 20, 20, 20

Figure 1.1

Lets rst nd the Nash equilibria for both cross sections and then we can determine theoverall Nash equilibria. Using the underline method, we obtain that the following are Nashequilibria when rooomate 1 plays 123 : (123; 213) and (213; 123) : When roommate 1 plays


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1.8. APPLICATION: POLITICAL ECONOMY 25

213; the Nash equilibrium is (123; 123) :


Roommate 2 123 20, 20, 20 18, 18, 24213 18, 24, 18 30, 15, 15


Roommate 2 123 24, 18, 18 15, 30, 15213 15, 15, 30 20, 20, 20

Figure 1.2

Lets now consider the Nash equilibria over all three roommates. When roommate 1 plays123; then players 2 and 3 optimally play (123; 213) : Does roommate 1 have an incentive todeviate, that is, to change her ranking to 213? Holding xed players 2 and 3 at (123; 213) ;roommate 1 currently has a payo of 18, while switching yields a payo of 15 (no deviation).When roommate 1 plays 123; then players 2 and 3 also optimally play (213; 123) : Doesroommate 1 have an incentive to deviate? Currently, roommate 1 has a payo of 18, whileswitching yields a payo of 15 (no deviation). When roommate 1 plays 213; then players 2and 3 optimally play (123; 123) : Does roommate 1 have an incentive to deviate? Currently,roommate 1 has a payo of 24, while switching yields a payo of 20 (no deviation).

Thus, the Nash equilibria are (123; 123; 213) ; (123; 213; 123) ; and (213; 123; 123) : In each,two players have ranking 123; while the third has ranking 213: Well return to this Room Se-lection Game when we discuss mixed-strategies in Chapter 3 and dynamic games in Chapter

4.

Exercise 1.4 Suppose that the payos for the rooms are now a = 20; b = 12; and c = 8 :What is the Nash equilibrium of the game?

1.8 Application: Political Economy

The next game considers the interaction between two citizens and two politicians (Obamaand Romney). Each citizen represents the political base of one of the two political parties(Democrat and Republican). We represent the political spectrum along the line from 1 to+1 : The left corresponds to the political left (Democrat), while the right corresponds to thepolitical right (Republican). All four of the players make simultaneous decisions.

The Democratic citizen prefers a platform at 12 ; with payo function given by12 m ;

where m is the platform of the elected politician. If the elected politician has platformm = 12 ; then the Democratic citizen has the highest payo value of 0: The Republican


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between Obama and Romney, as both have the same platform. Thus, the payo to Romneyis 12 ; as he has a 50% chance of winning. By voting, the Republican citizen has a payo of

12 0 =

12 from the dierence in platform minus another

14 as the cost of voting (for atotal payo of 34 ).

Suppose that the strategies are (Not Vote, 0). In this case, no one votes, meaning thatthe election is a tie. Each politician then has a 50% chance of winning for a payo of 12 :By not voting, the Republican citizen has a payo of 12 0 = 12 from the dierence inplatform.

Suppose that the strategies are (Vote, 38 ). In this case, only one citizen votes (theRepublican). The citizen votes for Romney, as Romneys platform is closer to 12 ; the bestplatform for the Republican citizen. The payo for Romney is then 1; as Romney wins theelection with certainty. By voting, the Republican citizen has a payo of 12 38 = 18from the dierence in platform minus another 14 as the cost of voting (for a total payo of

38 ).

Suppose that the strategies are (Not Vote, 38 ). In this case, no one votes, meaning that

the election is a tie. Each politician then has a 50% chance of winning for a payo of 12 : Bynot voting, the Republican citizen has a payo of (50%) 12 0 (50%) 12 38 = 516 fromthe expected dierence in platform. Recall that if Obama wins (50% chance), the platformis 0; while if Romney wins (50% chance), the platform is 38 :

Lets use the underline method for Figure 1.3.

Romney

0 3

8

Republican Vote 34 ; 12 38 ; 1citizen Not vote 12 ; 12 516 ; 12

The two Nash equilibria of this reduced game are (Not vote, 0) and (Not vote, 38 ). As (Notvote, 0) is a Nash equilibrium, given that we held xed the strategies of the other two playersat (Note vote, 0), then the strategies in Table 1.1 are in fact a Nash equilibrium.


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A second political equilibrium A second possible set of symmetric strategies is givenbelow.

Democrat citizen VoteRepublican citizen VoteObama platform 38Romney platform 38

Table 1.2

Lets hold xed the strategies for (Democratic citizen, Obama) at (Vote, 38 ). If we canshow the optimal strategies for (Republican citizen, Romney), holding xed the strategiesof the other two, are (Vote, 38 ), then the strategies in Table 1.2 are a Nash equilibrium.

Holding xed the strategies for (Democratic citizen, Obama) at (Vote,

3

8), the normal

form for (Republican citizen, Romney) is given below.

Romney0 38


Figure 1.4

Lets validate the payos for each of the four possible outcomes. Suppose that thestrategies are (Vote, 0). In this case, both citizens vote. The Democratic citizen votes forObama, while the Republican citizen votes for Romney. Thus, the payo to Romney is12 ; as he has a 50% chance of winning. By voting, the Republican citizen has a payo of

(50%) 12 38 (50%) 12 0 = 1116 from the expected dierence in platform minusanother 14 as the cost of voting (for a total payo of 1516 ). Recall that if Obama wins (50%chance), the platform is 38 ; while if Romney wins (50% chance), the platform is 0:

Suppose that the strategies are (Not Vote, 0). In this case, only the Democratic citizenvotes. This means that Obama wins the election with certainty. The payo to Romney isthen 0: By not voting, the Republican citizen has a payo of 12 38 = 78 from thedierence in platform.

Suppose that the strategies are (Vote, 38 ). In this case, both citizens vote. The Democraticcitizen votes for Obama, while the Republican citizen votes for Romney. Thus, the payo to Romney is 12 ; as he has a 50% chance of winning. By voting, the Republican citizen has apayo of (50%) 12 38 (50%) 12 38 = 12 from the expected dierence in platform


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minus another 14 as the cost of voting (for a total payo of 34 ).Suppose that the strategies are (Not Vote, 38 ). In this case, only the Democratic citizen

votes. This means that Obama wins the election with certainty. The payo to Romney isthen 0: By not voting, the Republican citizen has a payo of 12 38 = 78 from thedierence in platform.

Lets use the underline method for Figure 1.4.

Romney0 38


The two Nash equilibria of this reduced game are (Not vote, 0) and (Vote, 38 ). As (Vote,38 ) is a Nash equilibrium, given that we held xed the strategies of the other two players at(Vote, 38 ), then the strategies in Table 1.2 are a second Nash equilibrium.

Exercise 1.5 The town of Lakesville is located next to the town of West Lakesville. The Chinese restaurant Lins is located in Lakesville and the Chinese restaurant Wongs is located in West Lakesville. Each restaurant currently delivers take-out orders within its town only.Both restaurants are simultaneously deciding whether or not to expand their delivery service to the neighboring town (Lins to oer delivery to West Lakesville and Wongs to Lakesville).

A restaurant will earn $25 for selling take-out food in its own town and $15 for selling take-out food in the other town (a $10 travel cost is already included in the specied earnings).If a restaurant decides to expand its delivery service, a xed cost of $10 must be paid (to hire an additional driver).

If both expand their delivery service, they both maintain their current customers in their own town. If one expands and the other does not, then the one that expands sells to all consumers in both towns. The game is depicted below.

Wongs Expand Not expand

Lins Expand 15, 15 30, 0 Not expand 0, 30 25, 25

.

Find the Nash equilibria of this game.


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Exercise 1.6 In the previous problem, we assumed that consumers would make certain choices without actually including them in the game. Lets correct this. Suppose each town

only has one consumer of Chinese take-out food. Each consumer must select a Chinese restaurant. Exactly as in the previous problem, the restaurants must decide whether or not to expand delivery service. All decisions (both those of the restaurants and those of the consumers) are made simultaneously.

If a consumer selects a restaurant that does deliver take-out food to that consumer, then the consumer has a payo of 1. If a consumer selects a restaurant that does not deliver take-out food to that consumer, then no sale is made and the consumer has a payo of 0.

The payos for the restaurants are determined from the second paragraph of the previous exercise.

One of the following two sets of strategies is a Nash equilibrium. Which one is a Nash equilibrium? Justify your answer with sound reasoning.

Option 1 Option 2 Lins Restaurant Expand Wongs Restaurant Expand Lakesville consumer Select Lins West Lakesville consumer Select Wongs

Lins Restaurant Expand Wongs Restaurant Expand Lakesville consumer Select Wongs West Lakesville consumer Select Lins


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Chapter 2

Lunch Trucks at the Beach

2.1 Motivating Example

When you go to the beach, you probably enjoy the convenience of being able to obtain foodand beverage at a lunch truck conveniently parked on the beach. Have you ever wonderedhow these lunch trucks choose where they set up shop for the day? Lets consider the strategicdecisions of two lunch trucks in this Beach Game.

The beach is one mile long. Evenly distributed on the beach are consumers. We assumethat the consumers located immediately in front of a lunch truck will visit the lunch truck

with probability 1 (for a total demand of 1). The consumers located further away from thelunch truck will visit with probability less than 1:

Vendor 1 is initially located at mile 0 of the beach, but can costlessly move to any locationon the one-mile strip. Vendor 2 is initially located at mile 1 of the beach, but can costlesslymove to any location.

The two players in this game are the two vendors. Each must choose its location, withoutknowing the location chosen by the other vendor. Suppose that the equilibrium location forvendor 1 is n1 and the equilibrium location for vendor 2 is n2: If so, then the consumerlocated at mile z on the beach goes to vendor 1 if

jn

1z

j 30):The marginal cost of production can be either high chigh = 12 or low clow = 6 :(a) Two rms Let the market contain two rms, the rst with high marginal cost of production, c1 =

chigh = 12; and the second with low marginal cost of production, c2 = clow = 6 : Solve for the equilibrium output decisions of the two rms. What are the prots of each rm?

(b) 2n rms

Let the number of rms be 2n where n = 1 ; 2; :::; 1 : For the odd-numbered rms, the cost of production is high, ci = chigh = 12 for i = 1 ; 3; :::; 2n 1: For the even-numbered rms,the cost of production is low, ci = clow = 6 for i = 2 ; 4; :::; 2n: As n grows, do both high cost and low cost rms continue to produce (yes or no)? (Hint: the best response functions have a lower bound of q = 0). If one half of the rms stops producing, what is the value of n at which this happens? What is the output for each of the other half of the rms at this point? What is the equilibrium market price P at this point?

2.4 Bertrand Duopoly

The Bertrand model of rm competition has a dierent philosophy than the Cournot model.Rather than a competition in terms of quantity choices, the rms compete in terms of pricechoices. The consumers in the economy will purchase the good from the rm with thelowest price (with the purchases exactly split if both rms set the same price). We consideronly Bertrand duopolies in which two rms simultaneously select a price, where rm 1 hasmarginal cost of production equal to c1 and rm 2 has marginal cost of production equal toc2:

If c1 = c2; then the best responses for both rms are identical, with the best response forrm 1 given by:

p1 = b1 ( p2) = ( p2 if p2 > c 1c1 if p2 c1);where > 0 small. This function indicates that each rm wishes to undercut the price of theother rm, but never nds it optimal to set a price below its marginal cost (as this results


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38 CHAPTER 2. LUNCH TRUCKS AT THE BEACH

in negative prot). Thus, the unique Nash equilibrium is p1 = p2 = c1 = c2: Both rms, bypricing at their marginal costs, receive zero prot.

If c1 < c 2; then rm 2 will not be able to sell any output as rm 1 will simply set a priceequal to c2 ; for > 0 small.This basic Bertrand model seems limited. The predictions of the model are either (i)

both rms earn zero prot or (ii) one rm does not sell anything and goes out of business.To obtain a more realistic set of predictions, we incorporate transportation costs into themodel. Suppose that consumers are evenly distributed on the real line from 0 to 1: We canview this as a town that is organized along a 1-mile line. At the left end of town (mile 0),rm 1 operates a store. At the right end of town (mile 1), rm 2 operates a store. Eachconsumer incurs a cost of 1 per mile traveled. The consumers will travel to the store that

provides the lowest total cost, where the total cost is equal to price of the good plus thetransportation cost.

As with all Bertrand models, the two rms compete by simultaneously choosing prices p1 and p2: A consumer living at location z along the 1-mile strip goes to store 1 if p1 + z < p2 + (1 z ) and purchases one good (demand = 1). The same consumer goes to store 2 if p1 + z > p 2 + (1 z ) (again, demand = 1). If p1 + z = p2 + (1 z ); then the consumerat location z is indierent between the two stores and splits its demand. In particular, if p1 + z = p2 + (1 z ); then all consumers to the left of z go to store 1 and all consumers

to the right of z go to store 2. This is illustrated in Figure 2.2 for a given choice of ( p1; p2) :

Figure 2.2


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2.4. BERTRAND DUOPOLY 39

Assume that the consumers along the 1-mile strip have mass equal to 1 (and were previ-ously assumed to be evenly distributed). Thus, store 1 sells to consumers in the left t mile

of the town, which is equal to t consumers. Likewise, store 2 sells to consumers in the right(1 t ) mile, which is equal to (1 t) consumers.The payo function for store 1 is equal to

1 ( p1; p2) = ( p1 c1) t = ( p1 c1) p2 p1 + 1

2;

after using the denition of t from Figure 2.2. The rst order condition for store 1 is givenby:

@ 1 ( p1; p2)

@p1=

p2 p1 + 1

2

p1 c1

2= 0 :

Solving for p1; the best response for rm 1 is then given by p1 = b1 ( p2) = p2 +1+ c12 :The payo function and the rst order condition for rm 2 are given by:

2 ( p1; p2) = ( p2 c2) (1 t ) = ( p2 c2) p1 p2 + 1

2:

@ 2 ( p1; p2)@p2

= p1 p2 + 1

2 p2 c2

2= 0 :

The best response for rm 2 is p2 = b2( p1) = p1 +1+ c22 :The Nash equilibrium is ( p1; p2) such that p1 = b1 ( p2) and p1 = b2 ( p2) : Solving these

two equations (you can check my algebra) yields:

p1 = 1 + 13

c2 + 23

c1:

p2 = 1 + 13

c1 + 23

c2:

For the case in which c1 = c2; then both rms sell the good and do so at a price strictlygreater than marginal cost (which allows for strictly positive prot). This shows that adding

in a logical element like transportation costs allows for sensible predictions from the Bertrandmodel.

Exercise 2.4 Consider the Bertrand model of duopoly with transportation costs. Two rms compete in prices by simultaneously selecting prices p1 and p2: Firm 1 is located at mile 0 and rm 2 is located at mile 1. Consumers are uniformly distributed between mile 0 and mile 1 and incur a cost of 1 for each mile traveled to reach the selected rm. Consumers purchase


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from the rm with the lowest total cost (price plus travel cost) and have a demand equal to1:

The rms have marginal costs of production given by c1 and c2: The analysis in class was conducted under the implicit assumption that jc1 c2j 3: What happens if jc1 c2j > 3;say c1 = 8 and c2 = 4? Specically, what are the equilibrium price choices p1 and p2 and the prots of each rm?

2.5 Solution to the Motivating Example

Recall the Beach Game from the beginning of the chapter. Lets initially suppose that n1;the equilibrium location of vendor 1, and n2; the equilibrium location of vendor 2, are suchthat n1 n 2: If this turns out not to be true, then we would need to update our solution.

The payo for vendor 1 will be a function not just of its own choice, n1, but also of the choice of the other vendor, n2. Lets dene this payo function as p1 (n 1; n 2) : Hold-ing n2 xed, lets write an expression for the solution to vendor 1s optimization problemmax

0 n 1 1 p1 (n 1; n 2) : The solution to the problem is called the best response b1 (n 2) = argmax

0 n 1 1 p1 (n 1; n 2) : Likewise, the payo for vendor 2 is a function of both locations, so we de-ne p2 (n 1; n 2) : The solution to the problem is called the best response b2 (n 1) = arg max

0 n 2 1 p2 (n 1; n 2) : As a Nash equilibrium is a best response to a best response, then the Nashequilibrium (n 1; n 2) must solve the following two equations:

n 1 = b1 (n 2) :

n 2 = b2 (n 1) :

The diculty in the problem is then to dene p1 (n 1; n 2) and p2 (n 1; n 2) : The payo forany vendor is equal to the demand of the consumers that visit that particular truck (I havenot mentioned any costs for the trucks, so we assume that they dont have any). The totaldemand is equal to the area under the demand curve. Consider Figure 2.3.


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Figure 2.3

The payo functions are p1 (n 1; n 2) = area (A) + area (B ) and p2 (n 1; n 2) = area (C ) +area (D ): Using the geometric rule that the area of a quadrilateral is equal to average heighttimes average base, the areas of the four quadrilaterals in Figure 2.3 are given as follows.

area (A) = 1

2 [(1 n 1) + 1] (n 1)

area (B ) = 1

21

n 2 n 12

+ 1n 2 n 1

2

area (C ) = 1

21

n 2 n 12

+ 1n 2 n 1

2= area (B )

area (D ) = 1

2 (n 2 + 1)(1 n 2)

The areas can be equivalently written as

area (A) = n1 12 (n 1)2

area (B ) = area (C ) =n 2 n 1

2 12

n 2 n 12

2

area (D ) = 1

2 12

(n 2)2

Lets take the rst order condition of p1 (n 1; n 2) (the partial derivative of p1 (n 1; n 2) with


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respect to n1):@p1 (n 1; n 2)

@n1= 1 n 1

12

+ 12

n 2 n 12

= 0 :

Solving the equation for n1 :

54

n 1 = 1

2 +

14

n 2:

n 1 = 2

5 +

15

n 2:

This is the best response for vendor 1 (as a function of what vendor 2 does).Lets now take the rst order condition of p2 (n 1; n 2) :

@p2 (n 1; n 2)@n2

= 12 12 n 2 n 12 n 2 = 0 :

Solving the equation for n2 :

54

n 2 = 1

2 +

14

n 1:

n 2 = 2

5 +

15

n 1:

Solving the two equations in two unknowns yields the equilibrium locations (n 1; n 2) =12 ; 12 :

This is remarkably similar to the prediction of the Hotelling model of spatial competition.In the Hotelling model, the demand is constant as the distance increases (a citizen votes for acandidate if that candidate has the closest platform, meaning that the demand (vote number)is not a function of the distance from that platform to the citizens preference point). Justlike with candidates, the lunch trucks will continue to undercut each other until they choosethe exact same location (the median location).

Is this equilibrium optimal? The equilibrium payos for both are equal to p1 (n 1; n 2) = p

2 (n

1; n

2) = 3

8: Notice that if the vendors were to locate at (

n1; n

2) =1

4; 3

4; then the payo for each would be equal to p1 (n 1; n 2) = p2 (n 1; n 2) = 716 ; which is of course greater than the

equilibrium payo. This is the reality of allowing self-interested agents to make strategicdecisions: their decisions can often result in a suboptimal outcome.

Exercise 2.5 Consider the following update of the Beach Game. Vendor 1 is initially located at 0 and vendor 2 is initially located at 1: Both vendors incure a cost of k per mile for moving


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to a new location. Given the locations of the two vendors (n 1; n 2) ; the demand function of the consumers located at z along the beach (where 0 z 1) is dened as:

demand i (z ) = 8>:1 jn i z j if jn i z j< jn j z j

12 (1 jn i z j) if jn i z j= jn j z j

0 if jn i z j> jn j z j9>=>;

where (i; j ) = (1 ; 2) and (i; j ) = (2 ; 1) :Both vendors select their location simultaneously (cannot observe the location of the other

vendor prior to making their decision).As a function of k : 0 k 34 ; solve for the equilibrium vendor positions of this game.

Verify that the result obtained for the special case of k

= 0 is (n

1; n

2) =1

2; 1

2 (the answer obtained above for the zero cost case).


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Chapter 3

One of the Classic Blunders

3.1 Motivating Example

If you have ever seen the movie The Princess Bride, then you clearly remember this famousscene. The Dread Pirate Roberts has challenged the villain Vizzini to a "Battle of Wits."Two identical cups of wine are located on the table. Placing the two cups out of view of Vizzini, the Dread Pirate Roberts inserts the odorless, colorless poison "iocane powder" intoone cup. The Dread Pirate Roberts then places one cup on the table in front of himself and

the second on the table in front of Vizzini.The rst move is taken by the Dread Pirate Roberts, who must decide where to place

the cup containing the poison. The second move is to be taken by Vizzini, who now mustdecide which cup to drink from. Vizzini drinks from the cup that he chooses and the DreadPirate Roberts from the other. The choice by Vizzini is made without knowing the choicemade by the Dread Pirate Roberts. Thus, the game involves simultaneous choices that canbe analyzed using our game theoretic techniques.

Dene the payo from death (the poison is deadly) as 0 and the payo from living as 1:

The matrix form for this game is then given by:

VizziniCup near DPR Cup near Vizz

The Dread Poison near DPR 1, 0 0, 1Pirate Roberts Poison near Vizz 0, 1 1, 0

Example 3.1

.

45


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46 CHAPTER 3. ONE OF THE CLASSIC BLUNDERS

Lets consider the logic as quoted by Vizzini.

Vizzini: All I have to do is divine from what I know of you: are you the sortof man who would put the poison into his own goblet or his enemys? Now, aclever man would put the poison into his own goblet, because he would knowthat only a great fool would reach for what he was given. I am not a great fool,so I can clearly not choose the wine in front of you. But you must have knownI was not a great fool, you would have counted on it, so I can clearly not choosethe wine in front of me.

Are you following so far? What about the next logical deduction?

Vizzini: [The poison] iocane comes from Australia, as everyone knows, andAustralia is entirely peopled with criminals, and criminals are used to havingpeople not trust them, as you are not trusted by me, so I can clearly not choosethe wine in front of you. And you must have suspected I would have known thepowders origin, so I can clearly not choose the wine in front of me.

Vizzini then distracts the Dread Pirate Roberts and switches the cups on the table. Whenthe Dread Pirate Roberts returns his attention to the table, Vizzini chooses to drink from

the cup in front of him, meaning that the Dread Pirate Roberts must drink from the othercup. After drinking, the Dread Pirate Roberts informs Vizzini that he has lost. To this,Vizzini laughs and replies:

Vizzini: You only think I guessed wrong! Thats whats so funny! I switchedglasses when your back was turned! Ha ha! You fool! You fell victim to oneof the classic blunders - The most famous of which is "never get involved in aland war in Asia" - but only slightly less well-known is this: "Never go against aSicilian when death is on the line"!

At the end of the sentence, Vizzini falls to the ground dead. It turns out that the DreadPirate Roberts had placed the poison into both cups and had spent the past months buildingup an immunity to the poison. The moral of the story is never agree to a game which seemsto be a 50/50 gamble, because the person proposing the game must have more informationabout the game than you do. In this case, the Dread Pirate Roberts had changed the natureof the game from Example 3.1 into a game in which the he would always win.


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3.2. SOLVING THE BATTLE OF WITS 47

3.2 Solving the Battle of Wits

For our purposes, lets consider if there was an optimal way to play the game as viewed byVizzini (as given in Example 3.1). From the quotes by Vizzini, we see the problem withnding a Nash equilibrium. Each choice that Vizzini would make would lead the DreadPirate Roberts to make a dierent choice, which would then change the choice of Vizzini,and so forth. Using the underline method


The Dread Poison near DPR 1, 0 0, 1

Pirate Roberts Poison near Vizz 0, 1 1, 0

we see that a Nash equilibrium does not exist.

Up to this point, the Nash equilibria that we have been trying to nd are Nash equilibriain terms of pure strategies. When we said that (Fink, Fink) is a Nash equilibrium of thePrisoners Dilemma Game, we mean that each suspect is choosing Fink with probability100%. However, there is the possibility for the existence of another type of Nash equilibrium,a type that is called mixed-strategy Nash equilibrium. In a mixed-strategy Nash equilibrium,at least one of the players is dividing the 100% probability among multiple strategies. In

this case, the player is indierent between playing the strategies and uses a random numbergenerator to decide exactly which one is played.

A Nash equilibrium can be of one of two types: (i) pure-strategy Nash equilibrium (as dis-cussed in Chapters 1 and 2) and (ii) mixed-strategy Nash equilibrium (( p ; 1 p ) ; (q ; 1 q )) ;where player 1 assigns probability p to its rst strategy (with the remaining 1 p to itssecond) and player 2 assigns probability q to its rst strategy (with the remaining 1 q toits second). An important result in economics states that a Nash equilibrium must alwaysexist. This result was proven by John Nash and is beyond the scope of this class to discuss.If a pure-strategy Nash equilibrium does not exist (as is the case for Example 3.1), then amixed-strategy Nash equilibrium must exist.

If the Dread Pirate Roberts assigns probability p to the rst strategy and 1 p to thesecond, then he must be receiving the same expected payo from both strategies. This isbecause the expected payo for the Dread Pirate Roberts is inversely related to the expectedpayo of Vizzini. As one goes up, the other must necessarily go down. Payo maximizationdictates that the Dread Pirate Roberts will assign 100% probability to a strategy if its


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expected payo is strictly higher, resulting in a lower expected payo for Vizzini. Thus, inorder for Vizzini to maximize his own expected payo, he must set (q ; 1 q ) so that the

Dread Pirate Roberts is indierent between its two strategies.Lets write down how the expected payos are computed for the Dread Pirate Roberts.

We only consider the payos of the Dread Pirate Roberts:

VizziniCup near DPR ( q ) Cup near Vizz ( 1 q )

The Dread Poison near DPR 1 q ;::: 0 (1 q );:::Pirate Roberts Poison near Vizz 0 q ;::: 1 (1 q );:::

:

Using these expected payos, the value q (the probability chosen by Vizzini) is such thatthe Dread Pirate Roberts has the same expected payos for its two strategies (Poison nearDPR, Poison near Vizz):

1 q + 0 (1 q ) = 0 q + 1 (1 q ):

Algebra yields q = 12 :

Similarly, the best choice by the Dread Pirate Roberts is to choose the value p suchthat Vizzini is indierent between its two strategies. The expected payos are computed by

considering only the payos of Vizzini:


The Dread Poison near DPR ( p ) ..., 0 p ..., 1 pPirate Roberts Poison near Vizz ( 1 p ) ..., 1 (1 p ) ..., 0 (1 p )

:

Vizzini is indierent between its two strategies when

0 p + 1 (1 p ) = 1 p + 0 (1 p ):

Algebra yields p = 12 :

Thus, the unique Nash equilibrium is a mixed-strategy Nash equilibrium in which bothplayers perfectly randomize over their two strategies: (( p ; 1 p ) ; (q ; 1 q )) = 12 ;

12 ;

12 ;

12 :

As an alternative solution method, we can make a plot of the best response curves forboth players. Lets put the probabilities p on the x-axis and q on the y-axis. Given the


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Figure 3.2

A Nash equilibrium is dened as the intersection of bV izz and bDP R ; the best responsecurves for both players. As can be seen from Figure 3.2, the unique Nash equilibrium (anypure-strategy Nash equilibria would appear as intersections on the boundary) is given by( p ; q ) = 12 ;

12 ; which is exactly what we found from the algebra above.

3.3 Solving Similar Games

As another example, lets consider a pivotal play in soccer: the penalty kick. The penaltykick is a strategic interaction between a Scorer and a Goalie. The Scorer is located a mere 12yards from the Goalie, who is positioned on the goal line. The Scorer decides either to kickthe ball to his left (Shoot Left) or kick the ball to his right (Shoot Right). Given the sizeof the soccer goal and the speed at which a soccer ball travels, the only way for a Goalie toblock a well-placed shot is to dive before knowing which direction the shot is taken. Thus,without knowing the actions of the Scorer, the Goalie can either dive to his left (Dive Left)or dive to his right (Di

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